Advertisement

Elliptic Feynman integrals and pure functions

  • Johannes Broedel
  • Claude Duhr
  • Falko Dulat
  • Brenda Penante
  • Lorenzo TancrediEmail author
Open Access
Regular Article - Theoretical Physics
  • 14 Downloads

Abstract

We propose a variant of elliptic multiple polylogarithms that have at most logarithmic singularities in all variables and satisfy a differential equation without homogeneous term. We investigate several non-trivial elliptic two-loop Feynman integrals with up to three external legs and express them in terms of our functions. We observe that in all cases they evaluate to pure combinations of elliptic multiple polylogarithms of uniform weight. This is the first time that a notion of uniform weight is observed in the context of Feynman integrals that evaluate to elliptic polylogarithms.

Keywords

NLO Computations QCD Phenomenology 

Notes

Open Access

This article is distributed under the terms of the Creative Commons Attribution License (CC-BY 4.0), which permits any use, distribution and reproduction in any medium, provided the original author(s) and source are credited.

References

  1. [1]
    A.B. Goncharov, Geometry of configurations, polylogarithms, and motivic cohomology, Adv. Math. 114 (1995) 197.MathSciNetCrossRefzbMATHGoogle Scholar
  2. [2]
    E. Remiddi and J.A.M. Vermaseren, Harmonic polylogarithms, Int. J. Mod. Phys. A 15 (2000) 725 [hep-ph/9905237] [INSPIRE].
  3. [3]
    A.B. Goncharov, Multiple polylogarithms, cyclotomy and modular complexes, Math. Res. Lett. 5 (1998) 497 [arXiv:1105.2076] [INSPIRE].MathSciNetCrossRefzbMATHGoogle Scholar
  4. [4]
    J. Vollinga and S. Weinzierl, Numerical evaluation of multiple polylogarithms, Comput. Phys. Commun. 167 (2005) 177 [hep-ph/0410259] [INSPIRE].
  5. [5]
    A.B. Goncharov, M. Spradlin, C. Vergu and A. Volovich, Classical Polylogarithms for Amplitudes and Wilson Loops, Phys. Rev. Lett. 105 (2010) 151605 [arXiv:1006.5703] [INSPIRE].
  6. [6]
    A.B. Goncharov, A simple construction of Grassmannian polylogarithms, arXiv:0908.2238 [INSPIRE].
  7. [7]
    K.T. Chen, Iterated path integrals, Bull. Am. Math. Soc. 83 (1977) 831 [INSPIRE].MathSciNetCrossRefzbMATHGoogle Scholar
  8. [8]
    F.C.S. Brown, Multiple zeta values and periods of moduli spaces \( {\overline{\mathfrak{M}}}_{0,n} \), Annales Sci. Ecole Norm. Sup. 42 (2009) 371 [math.AG/0606419] [INSPIRE].
  9. [9]
    C. Duhr, H. Gangl and J.R. Rhodes, From polygons and symbols to polylogarithmic functions, JHEP 10 (2012) 075 [arXiv:1110.0458] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  10. [10]
    J. Ablinger, J. Blümlein and C. Schneider, Harmonic Sums and Polylogarithms Generated by Cyclotomic Polynomials, J. Math. Phys. 52 (2011) 102301 [arXiv:1105.6063] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  11. [11]
    S. Buehler and C. Duhr, CHAPLIN — Complex Harmonic Polylogarithms in Fortran, Comput. Phys. Commun. 185 (2014) 2703 [arXiv:1106.5739] [INSPIRE].ADSCrossRefzbMATHGoogle Scholar
  12. [12]
    C. Duhr, Hopf algebras, coproducts and symbols: an application to Higgs boson amplitudes, JHEP 08 (2012) 043 [arXiv:1203.0454] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  13. [13]
    C. Anastasiou, C. Duhr, F. Dulat and B. Mistlberger, Soft triple-real radiation for Higgs production at N 3LO, JHEP 07 (2013) 003 [arXiv:1302.4379] [INSPIRE].
  14. [14]
    J.M. Henn, Multiloop integrals in dimensional regularization made simple, Phys. Rev. Lett. 110 (2013) 251601 [arXiv:1304.1806] [INSPIRE].ADSCrossRefGoogle Scholar
  15. [15]
    J. Ablinger, J. Blümlein and C. Schneider, Analytic and Algorithmic Aspects of Generalized Harmonic Sums and Polylogarithms, J. Math. Phys. 54 (2013) 082301 [arXiv:1302.0378] [INSPIRE].
  16. [16]
    E. Panzer, Algorithms for the symbolic integration of hyperlogarithms with applications to Feynman integrals, Comput. Phys. Commun. 188 (2015) 148 [arXiv:1403.3385] [INSPIRE].ADSCrossRefzbMATHGoogle Scholar
  17. [17]
    C. Bogner and F.C.S. Brown, Feynman integrals and iterated integrals on moduli spaces of curves of genus zero, Commun. Num. Theor. Phys. 09 (2015) 189 [arXiv:1408.1862] [INSPIRE].MathSciNetCrossRefzbMATHGoogle Scholar
  18. [18]
    C. Duhr, Mathematical aspects of scattering amplitudes, in proceedings of the Theoretical Advanced Study Institute in Elementary Particle Physics: Journeys Through the Precision Frontier: Amplitudes for Colliders. (TASI 2014), Boulder, Colorado, U.S.A., 2-27 June 2014, pp. 419-476 [arXiv:1411.7538] [ https://doi.org/10.1142/9789814678766_0010] [INSPIRE].
  19. [19]
    J. Ablinger, J. Blümlein, C.G. Raab and C. Schneider, Iterated Binomial Sums and their Associated Iterated Integrals, J. Math. Phys. 55 (2014) 112301 [arXiv:1407.1822] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  20. [20]
    J.M. Henn, Lectures on differential equations for Feynman integrals, J. Phys. A 48 (2015) 153001 [arXiv:1412.2296] [INSPIRE].
  21. [21]
    H. Frellesvig, D. Tommasini and C. Wever, On the reduction of generalized polylogarithms to Lin and Li2,2 and on the evaluation thereof, JHEP 03 (2016) 189 [arXiv:1601.02649] [INSPIRE].
  22. [22]
    H. Frellesvig, Generalized Polylogarithms in Maple, arXiv:1806.02883 [INSPIRE].
  23. [23]
    N. Arkani-Hamed, J.L. Bourjaily, F. Cachazo and J. Trnka, Local Integrals for Planar Scattering Amplitudes, JHEP 06 (2012) 125 [arXiv:1012.6032] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  24. [24]
    A.V. Kotikov, Differential equations method: New technique for massive Feynman diagrams calculation, Phys. Lett. B 254 (1991) 158 [INSPIRE].
  25. [25]
    E. Remiddi, Differential equations for Feynman graph amplitudes, Nuovo Cim. A 110 (1997) 1435 [hep-th/9711188] [INSPIRE].
  26. [26]
    T. Gehrmann and E. Remiddi, Differential equations for two loop four point functions, Nucl. Phys. B 580 (2000) 485 [hep-ph/9912329] [INSPIRE].
  27. [27]
    J.M. Henn, A.V. Smirnov and V.A. Smirnov, Analytic results for planar three-loop four-point integrals from a Knizhnik-Zamolodchikov equation, JHEP 07 (2013) 128 [arXiv:1306.2799] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  28. [28]
    T. Gehrmann, L. Tancredi and E. Weihs, Two-loop master integrals for \( q\overline{q}\to V\ V \) : the planar topologies, JHEP 08 (2013) 070 [arXiv:1306.6344] [INSPIRE].
  29. [29]
    T. Gehrmann, A. von Manteuffel, L. Tancredi and E. Weihs, The two-loop master integrals for \( q\overline{q}\to V\ V \), JHEP 06 (2014) 032 [arXiv:1404.4853] [INSPIRE].
  30. [30]
    J.M. Henn and V.A. Smirnov, Analytic results for two-loop master integrals for Bhabha scattering I, JHEP 11 (2013) 041 [arXiv:1307.4083] [INSPIRE].ADSCrossRefGoogle Scholar
  31. [31]
    J.M. Henn, A.V. Smirnov and V.A. Smirnov, Evaluating single-scale and/or non-planar diagrams by differential equations, JHEP 03 (2014) 088 [arXiv:1312.2588] [INSPIRE].ADSCrossRefGoogle Scholar
  32. [32]
    S. Di Vita, P. Mastrolia, U. Schubert and V. Yundin, Three-loop master integrals for ladder-box diagrams with one massive leg, JHEP 09 (2014) 148 [arXiv:1408.3107] [INSPIRE].ADSCrossRefGoogle Scholar
  33. [33]
    J.M. Henn, K. Melnikov and V.A. Smirnov, Two-loop planar master integrals for the production of off-shell vector bosons in hadron collisions, JHEP 05 (2014) 090 [arXiv:1402.7078] [INSPIRE].ADSCrossRefGoogle Scholar
  34. [34]
    S. Caron-Huot and J.M. Henn, Iterative structure of finite loop integrals, JHEP 06 (2014) 114 [arXiv:1404.2922] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  35. [35]
    F. Caola, J.M. Henn, K. Melnikov and V.A. Smirnov, Non-planar master integrals for the production of two off-shell vector bosons in collisions of massless partons, JHEP 09 (2014) 043 [arXiv:1404.5590] [INSPIRE].ADSCrossRefGoogle Scholar
  36. [36]
    A. Grozin, J.M. Henn, G.P. Korchemsky and P. Marquard, Three Loop Cusp Anomalous Dimension in QCD, Phys. Rev. Lett. 114 (2015) 062006 [arXiv:1409.0023] [INSPIRE].
  37. [37]
    R. Bonciani, V. Del Duca, H. Frellesvig, J.M. Henn, F. Moriello and V.A. Smirnov, Next-to-leading order QCD corrections to the decay width H, JHEP 08 (2015) 108 [arXiv:1505.00567] [INSPIRE].
  38. [38]
    A. Grozin, J.M. Henn, G.P. Korchemsky and P. Marquard, The three-loop cusp anomalous dimension in QCD and its supersymmetric extensions, JHEP 01 (2016) 140 [arXiv:1510.07803] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  39. [39]
    T. Gehrmann, J.M. Henn and N.A. Lo Presti, Analytic form of the two-loop planar five-gluon all-plus-helicity amplitude in QCD, Phys. Rev. Lett. 116 (2016) 062001 [Erratum ibid. 116 (2016) 189903] [arXiv:1511.05409] [INSPIRE].
  40. [40]
    J.M. Henn, A.V. Smirnov, V.A. Smirnov and M. Steinhauser, A planar four-loop form factor and cusp anomalous dimension in QCD, JHEP 05 (2016) 066 [arXiv:1604.03126] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  41. [41]
    J.M. Henn and B. Mistlberger, Four-Gluon Scattering at Three Loops, Infrared Structure and the Regge Limit, Phys. Rev. Lett. 117 (2016) 171601 [arXiv:1608.00850] [INSPIRE].ADSCrossRefGoogle Scholar
  42. [42]
    J.M. Henn, A.V. Smirnov and V.A. Smirnov, Analytic results for planar three-loop integrals for massive form factors, JHEP 12 (2016) 144 [arXiv:1611.06523] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  43. [43]
    J.M. Henn, A.V. Smirnov, V.A. Smirnov, M. Steinhauser and R.N. Lee, Four-loop photon quark form factor and cusp anomalous dimension in the large-N c limit of QCD, JHEP 03 (2017) 139 [arXiv:1612.04389] [INSPIRE].
  44. [44]
    R. Bonciani, S. Di Vita, P. Mastrolia and U. Schubert, Two-Loop Master Integrals for the mixed EW-QCD virtual corrections to Drell-Yan scattering, JHEP 09 (2016) 091 [arXiv:1604.08581] [INSPIRE].ADSCrossRefGoogle Scholar
  45. [45]
    S. Di Vita, P. Mastrolia, A. Primo and U. Schubert, Two-loop master integrals for the leading QCD corrections to the Higgs coupling to a W pair and to the triple gauge couplings ZWW and γ WW, JHEP 04 2017) 008 [arXiv:1702.07331] [INSPIRE].
  46. [46]
    P. Mastrolia, M. Passera, A. Primo and U. Schubert, Master integrals for the NNLO virtual corrections to μe scattering in QED: the planar graphs, JHEP 11 (2017) 198 [arXiv:1709.07435] [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  47. [47]
    S. Di Vita, S. Laporta, P. Mastrolia, A. Primo and U. Schubert, Master integrals for the NNLO virtual corrections to μe scattering in QED: the non-planar graphs, JHEP 09 (2018) 016 [arXiv:1806.08241] [INSPIRE].CrossRefGoogle Scholar
  48. [48]
    T. Gehrmann, J.M. Henn and N.A. Lo Presti, Pentagon functions for massless planar scattering amplitudes, JHEP 10 (2018) 103 [arXiv:1807.09812] [INSPIRE].ADSCrossRefzbMATHGoogle Scholar
  49. [49]
    D. Chicherin, T. Gehrmann, J.M. Henn, N.A. Lo Presti, V. Mitev and P. Wasser, Analytic result for the nonplanar hexa-box integrals, arXiv:1809.06240 [INSPIRE].
  50. [50]
    Z. Bern, L.J. Dixon, D.C. Dunbar and D.A. Kosower, One loop n point gauge theory amplitudes, unitarity and collinear limits, Nucl. Phys. B 425 (1994) 217 [hep-ph/9403226] [INSPIRE].
  51. [51]
    Z. Bern, L.J. Dixon, D.C. Dunbar and D.A. Kosower, Fusing gauge theory tree amplitudes into loop amplitudes, Nucl. Phys. B 435 (1995) 59 [hep-ph/9409265] [INSPIRE].
  52. [52]
    Z. Bern, J.S. Rozowsky and B. Yan, Two loop four gluon amplitudes in N = 4 super-Yang-Mills, Phys. Lett. B 401 (1997) 273 [hep-ph/9702424] [INSPIRE].
  53. [53]
    Z. Bern, V. Del Duca, L.J. Dixon and D.A. Kosower, All non-maximally-helicity-violating one-loop seven-gluon amplitudes in N = 4 super-Yang-Mills theory, Phys. Rev. D 71 (2005) 045006 [hep-th/0410224] [INSPIRE].
  54. [54]
    C. Anastasiou, Z. Bern, L.J. Dixon and D.A. Kosower, Planar amplitudes in maximally supersymmetric Yang-Mills theory, Phys. Rev. Lett. 91 (2003) 251602 [hep-th/0309040] [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  55. [55]
    Z. Bern, L.J. Dixon and D.A. Kosower, All Next-to-maximally-helicity-violating one-loop gluon amplitudes in N = 4 super-Yang-Mills theory, Phys. Rev. D 72 (2005) 045014 [hep-th/0412210] [INSPIRE].
  56. [56]
    Z. Bern, L.J. Dixon and V.A. Smirnov, Iteration of planar amplitudes in maximally supersymmetric Yang-Mills theory at three loops and beyond, Phys. Rev. D 72 (2005) 085001 [hep-th/0505205] [INSPIRE].
  57. [57]
    V. Del Duca, C. Duhr and V.A. Smirnov, An Analytic Result for the Two-Loop Hexagon Wilson Loop in N = 4 SYM, JHEP 03 (2010) 099 [arXiv:0911.5332] [INSPIRE].
  58. [58]
    V. Del Duca, C. Duhr and V.A. Smirnov, The Two-Loop Hexagon Wilson Loop in N = 4 SYM, JHEP 05 (2010) 084 [arXiv:1003.1702] [INSPIRE].
  59. [59]
    L.J. Dixon, J.M. Drummond and J.M. Henn, Analytic result for the two-loop six-point NMHV amplitude in N = 4 super Yang-Mills theory, JHEP 01 (2012) 024 [arXiv:1111.1704] [INSPIRE].
  60. [60]
    L.J. Dixon, J.M. Drummond and J.M. Henn, Bootstrapping the three-loop hexagon, JHEP 11 (2011) 023 [arXiv:1108.4461] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  61. [61]
    L.J. Dixon, C. Duhr and J. Pennington, Single-valued harmonic polylogarithms and the multi-Regge limit, JHEP 10 (2012) 074 [arXiv:1207.0186] [INSPIRE].ADSCrossRefGoogle Scholar
  62. [62]
    L.J. Dixon, J.M. Drummond, M. von Hippel and J. Pennington, Hexagon functions and the three-loop remainder function, JHEP 12 (2013) 049 [arXiv:1308.2276] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  63. [63]
    L.J. Dixon and M. von Hippel, Bootstrapping an NMHV amplitude through three loops, JHEP 10 (2014) 065 [arXiv:1408.1505] [INSPIRE].ADSCrossRefGoogle Scholar
  64. [64]
    L.J. Dixon, J.M. Drummond, C. Duhr and J. Pennington, The four-loop remainder function and multi-Regge behavior at NNLLA in planar N = 4 super-Yang-Mills theory, JHEP 06 (2014) 116 [arXiv:1402.3300] [INSPIRE].
  65. [65]
    J.M. Drummond, G. Papathanasiou and M. Spradlin, A Symbol of Uniqueness: The Cluster Bootstrap for the 3-Loop MHV Heptagon, JHEP 03 (2015) 072 [arXiv:1412.3763] [INSPIRE].
  66. [66]
    L.J. Dixon, M. von Hippel and A.J. McLeod, The four-loop six-gluon NMHV ratio function, JHEP 01 (2016) 053 [arXiv:1509.08127] [INSPIRE].CrossRefGoogle Scholar
  67. [67]
    S. Caron-Huot, L.J. Dixon, A. McLeod and M. von Hippel, Bootstrapping a Five-Loop Amplitude Using Steinmann Relations, Phys. Rev. Lett. 117 (2016) 241601 [arXiv:1609.00669] [INSPIRE].ADSCrossRefGoogle Scholar
  68. [68]
    L.J. Dixon, J. Drummond, T. Harrington, A.J. McLeod, G. Papathanasiou and M. Spradlin, Heptagons from the Steinmann Cluster Bootstrap, JHEP 02 (2017) 137 [arXiv:1612.08976] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  69. [69]
    V. Del Duca et al., Multi-Regge kinematics and the moduli space of Riemann spheres with marked points, JHEP 08 (2016) 152 [arXiv:1606.08807] [INSPIRE].
  70. [70]
    V. Del Duca et al., The seven-gluon amplitude in multi-Regge kinematics beyond leading logarithmic accuracy, JHEP 06 (2018) 116 [arXiv:1801.10605] [INSPIRE].
  71. [71]
    A.V. Kotikov and L.N. Lipatov, DGLAP and BFKL evolution equations in the N = 4 supersymmetric gauge theory, in proceedings of the 35th Annual Winter School on Nuclear and Particle Physics, Repino, Russia, 19-25 February 2001, hep-ph/0112346 [INSPIRE].
  72. [72]
    A.V. Kotikov and L.N. Lipatov, DGLAP and BFKL equations in the N = 4 supersymmetric gauge theory, Nucl. Phys. B 661 (2003) 19 [Erratum ibid. B 685 (2004) 405] [hep-ph/0208220] [INSPIRE].
  73. [73]
    A.V. Kotikov, L.N. Lipatov, A.I. Onishchenko and V.N. Velizhanin, Three loop universal anomalous dimension of the Wilson operators in N = 4 SUSY Yang-Mills model, Phys. Lett. B 595 (2004) 521 [Erratum ibid. B 632 (2006) 754] [hep-th/0404092] [INSPIRE].
  74. [74]
    A.V. Kotikov, L.N. Lipatov, A. Rej, M. Staudacher and V.N. Velizhanin, Dressing and wrapping, J. Stat. Mech. 0710 (2007) P10003 [arXiv:0704.3586] [INSPIRE].
  75. [75]
    N. Beisert, B. Eden and M. Staudacher, Transcendentality and Crossing, J. Stat. Mech. 0701 (2007) P01021 [hep-th/0610251] [INSPIRE].
  76. [76]
    Ø. Almelid, C. Duhr and E. Gardi, Three-loop corrections to the soft anomalous dimension in multileg scattering, Phys. Rev. Lett. 117 (2016) 172002 [arXiv:1507.00047] [INSPIRE].ADSCrossRefGoogle Scholar
  77. [77]
    L.J. Dixon, The Principle of Maximal Transcendentality and the Four-Loop Collinear Anomalous Dimension, JHEP 01 (2018) 075 [arXiv:1712.07274] [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  78. [78]
    W.L. van Neerven, Infrared Behavior of On-shell Form-factors in a N = 4 Supersymmetric Yang-Mills Field Theory, Z. Phys. C 30 (1986) 595 [INSPIRE].
  79. [79]
    T. Gehrmann, J.M. Henn and T. Huber, The three-loop form factor in N = 4 super Yang-Mills, JHEP 03 (2012) 101 [arXiv:1112.4524] [INSPIRE].
  80. [80]
    A. Brandhuber, G. Travaglini and G. Yang, Analytic two-loop form factors in N = 4 SYM, JHEP 05 (2012) 082 [arXiv:1201.4170] [INSPIRE].
  81. [81]
    A. Brandhuber, B. Penante, G. Travaglini and C. Wen, The last of the simple remainders, JHEP 08 (2014) 100 [arXiv:1406.1443] [INSPIRE].ADSCrossRefGoogle Scholar
  82. [82]
    B. Eden, P.S. Howe, C. Schubert, E. Sokatchev and P.C. West, Four point functions in N = 4 supersymmetric Yang-Mills theory at two loops, Nucl. Phys. B 557 (1999) 355 [hep-th/9811172] [INSPIRE].
  83. [83]
    F. Gonzalez-Rey, I.Y. Park and K. Schalm, A Note on four point functions of conformal operators in N = 4 superYang-Mills, Phys. Lett. B 448 (1999) 37 [hep-th/9811155] [INSPIRE].
  84. [84]
    B. Eden, C. Schubert and E. Sokatchev, Three loop four point correlator in N = 4 SYM, Phys. Lett. B 482 (2000) 309 [hep-th/0003096] [INSPIRE].
  85. [85]
    M. Bianchi, S. Kovacs, G. Rossi and Y.S. Stanev, Anomalous dimensions in N = 4 SYM theory at order g 4, Nucl. Phys. B 584 (2000) 216 [hep-th/0003203] [INSPIRE].
  86. [86]
    J. Drummond, C. Duhr, B. Eden, P. Heslop, J. Pennington and V.A. Smirnov, Leading singularities and off-shell conformal integrals, JHEP 08 (2013) 133 [arXiv:1303.6909] [INSPIRE].
  87. [87]
    Y. Li, A. von Manteuffel, R.M. Schabinger and H.X. Zhu, N 3 LO Higgs boson and Drell-Yan production at threshold: The one-loop two-emission contribution, Phys. Rev. D 90 (2014) 053006 [arXiv:1404.5839] [INSPIRE].
  88. [88]
    Y. Li, A. von Manteuffel, R.M. Schabinger and H.X. Zhu, Soft-virtual corrections to Higgs production at N 3 LO, Phys. Rev. D 91 (2015) 036008 [arXiv:1412.2771] [INSPIRE].
  89. [89]
    A. Sabry, Fourth order spectral functions for the electron propagator, Nucl. Phys. 33 (1962) 401.MathSciNetCrossRefzbMATHGoogle Scholar
  90. [90]
    D.J. Broadhurst, The Master Two Loop Diagram With Masses, Z. Phys. C 47 (1990) 115 [INSPIRE].
  91. [91]
    S. Bauberger, F.A. Berends, M. Böhm and M. Buza, Analytical and numerical methods for massive two loop selfenergy diagrams, Nucl. Phys. B 434 (1995) 383 [hep-ph/9409388] [INSPIRE].
  92. [92]
    S. Bauberger and M. Böhm, Simple one-dimensional integral representations for two loop selfenergies: The Master diagram, Nucl. Phys. B 445 (1995) 25 [hep-ph/9501201] [INSPIRE].
  93. [93]
    S. Laporta and E. Remiddi, Analytic treatment of the two loop equal mass sunrise graph, Nucl. Phys. B 704 (2005) 349 [hep-ph/0406160] [INSPIRE].
  94. [94]
    B.A. Kniehl, A.V. Kotikov, A. Onishchenko and O. Veretin, Two-loop sunset diagrams with three massive lines, Nucl. Phys. B 738 (2006) 306 [hep-ph/0510235] [INSPIRE].
  95. [95]
    U. Aglietti, R. Bonciani, L. Grassi and E. Remiddi, The Two loop crossed ladder vertex diagram with two massive exchanges, Nucl. Phys. B 789 (2008) 45 [arXiv:0705.2616] [INSPIRE].
  96. [96]
    M. Czakon and A. Mitov, Inclusive Heavy Flavor Hadroproduction in NLO QCD: The Exact Analytic Result, Nucl. Phys. B 824 (2010) 111 [arXiv:0811.4119] [INSPIRE].
  97. [97]
    S. Caron-Huot and K.J. Larsen, Uniqueness of two-loop master contours, JHEP 10 (2012) 026 [arXiv:1205.0801] [INSPIRE].ADSCrossRefGoogle Scholar
  98. [98]
    D. Nandan, M.F. Paulos, M. Spradlin and A. Volovich, Star Integrals, Convolutions and Simplices, JHEP 05 (2013) 105 [arXiv:1301.2500] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  99. [99]
    S. Bloch and P. Vanhove, The elliptic dilogarithm for the sunset graph, J. Number Theor. 148 (2015) 328 [arXiv:1309.5865] [INSPIRE].MathSciNetCrossRefzbMATHGoogle Scholar
  100. [100]
    F.C.S. Brown and O. Schnetz, Modular forms in Quantum Field Theory, Commun. Num. Theor. Phys. 07 (2013) 293 [arXiv:1304.5342] [INSPIRE].MathSciNetCrossRefzbMATHGoogle Scholar
  101. [101]
    E. Remiddi and L. Tancredi, Schouten identities for Feynman graph amplitudes; The Master Integrals for the two-loop massive sunrise graph, Nucl. Phys. B 880 (2014) 343 [arXiv:1311.3342] [INSPIRE].
  102. [102]
    L. Adams, C. Bogner and S. Weinzierl, The two-loop sunrise graph with arbitrary masses, J. Math. Phys. 54 (2013) 052303 [arXiv:1302.7004] [INSPIRE].
  103. [103]
    R. Huang and Y. Zhang, On Genera of Curves from High-loop Generalized Unitarity Cuts, JHEP 04 (2013) 080 [arXiv:1302.1023] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  104. [104]
    J.D. Hauenstein, R. Huang, D. Mehta and Y. Zhang, Global Structure of Curves from Generalized Unitarity Cut of Three-loop Diagrams, JHEP 02 (2015) 136 [arXiv:1408.3355] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  105. [105]
    M. Søgaard and Y. Zhang, Elliptic Functions and Maximal Unitarity, Phys. Rev. D 91 (2015) 081701 [arXiv:1412.5577] [INSPIRE].
  106. [106]
    L. Adams, C. Bogner and S. Weinzierl, The two-loop sunrise graph in two space-time dimensions with arbitrary masses in terms of elliptic dilogarithms, J. Math. Phys. 55 (2014) 102301 [arXiv:1405.5640] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  107. [107]
    L. Adams, C. Bogner and S. Weinzierl, The two-loop sunrise integral around four space-time dimensions and generalisations of the Clausen and Glaisher functions towards the elliptic case, J. Math. Phys. 56 (2015) 072303 [arXiv:1504.03255] [INSPIRE].
  108. [108]
    L. Adams, C. Bogner and S. Weinzierl, The iterated structure of the all-order result for the two-loop sunrise integral, J. Math. Phys. 57 (2016) 032304 [arXiv:1512.05630] [INSPIRE].
  109. [109]
    A. Georgoudis and Y. Zhang, Two-loop Integral Reduction from Elliptic and Hyperelliptic Curves, JHEP 12 (2015) 086 [arXiv:1507.06310] [INSPIRE].ADSMathSciNetzbMATHGoogle Scholar
  110. [110]
    E. Remiddi and L. Tancredi, Differential equations and dispersion relations for Feynman amplitudes. The two-loop massive sunrise and the kite integral, Nucl. Phys. B 907 (2016) 400 [arXiv:1602.01481] [INSPIRE].
  111. [111]
    R. Bonciani, V. Del Duca, H. Frellesvig, J.M. Henn, F. Moriello and V.A. Smirnov, Two-loop planar master integrals for Higgs → 3 partons with full heavy-quark mass dependence, JHEP 12 (2016) 096 [arXiv:1609.06685] [INSPIRE].
  112. [112]
    S. Bloch, M. Kerr and P. Vanhove, Local mirror symmetry and the sunset Feynman integral, Adv. Theor. Math. Phys. 21 (2017) 1373 [arXiv:1601.08181] [INSPIRE].MathSciNetCrossRefzbMATHGoogle Scholar
  113. [113]
    L. Adams, C. Bogner, A. Schweitzer and S. Weinzierl, The kite integral to all orders in terms of elliptic polylogarithms, J. Math. Phys. 57 (2016) 122302 [arXiv:1607.01571] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  114. [114]
    G. Passarino, Elliptic Polylogarithms and Basic Hypergeometric Functions, Eur. Phys. J. C 77 (2017) 77 [arXiv:1610.06207] [INSPIRE].
  115. [115]
    J. Broedel, C. Duhr, F. Dulat and L. Tancredi, Elliptic polylogarithms and iterated integrals on elliptic curves II: an application to the sunrise integral, Phys. Rev. D 97 (2018) 116009 [arXiv:1712.07095] [INSPIRE].
  116. [116]
    A. von Manteuffel and L. Tancredi, A non-planar two-loop three-point function beyond multiple polylogarithms, JHEP 06 (2017) 127 [arXiv:1701.05905] [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  117. [117]
    A. Primo and L. Tancredi, Maximal cuts and differential equations for Feynman integrals. An application to the three-loop massive banana graph, Nucl. Phys. B 921 (2017) 316 [arXiv:1704.05465] [INSPIRE].
  118. [118]
    J. Ablinger et al., Iterated Elliptic and Hypergeometric Integrals for Feynman Diagrams, J. Math. Phys. 59 (2018) 062305 [arXiv:1706.01299] [INSPIRE].
  119. [119]
    L.-B. Chen, Y. Liang and C.-F. Qiao, NNLO QCD corrections to γ + η c(η b) exclusive production in electron-positron collision, JHEP 01 (2018) 091 [arXiv:1710.07865] [INSPIRE].
  120. [120]
    J.L. Bourjaily, A.J. McLeod, M. Spradlin, M. von Hippel and M. Wilhelm, Elliptic Double-Box Integrals: Massless Scattering Amplitudes beyond Polylogarithms, Phys. Rev. Lett. 120 (2018) 121603 [arXiv:1712.02785] [INSPIRE].
  121. [121]
    L.-B. Chen, J. Jiang and C.-F. Qiao, Two-Loop integrals for CP-even heavy quarkonium production and decays: Elliptic Sectors, JHEP 04 (2018) 080 [arXiv:1712.03516] [INSPIRE].ADSCrossRefGoogle Scholar
  122. [122]
    J. Broedel, C. Duhr, F. Dulat and L. Tancredi, Elliptic polylogarithms and iterated integrals on elliptic curves. Part I: general formalism, JHEP 05 (2018) 093 [arXiv:1712.07089] [INSPIRE].
  123. [123]
    M. Hidding and F. Moriello, All orders structure and efficient computation of linearly reducible elliptic Feynman integrals, arXiv:1712.04441 [INSPIRE].
  124. [124]
    M. Becchetti and R. Bonciani, Two-Loop Master Integrals for the Planar QCD Massive Corrections to Di-photon and Di-jet Hadro-production, JHEP 01 (2018) 048 [arXiv:1712.02537] [INSPIRE].ADSCrossRefGoogle Scholar
  125. [125]
    B. Mistlberger, Higgs boson production at hadron colliders at N 3 LO in QCD, JHEP 05 (2018) 028 [arXiv:1802.00833] [INSPIRE].
  126. [126]
    J. Broedel, C. Duhr, F. Dulat, B. Penante and L. Tancredi, Elliptic symbol calculus: from elliptic polylogarithms to iterated integrals of Eisenstein series, JHEP 08 (2018) 014 [arXiv:1803.10256] [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  127. [127]
    L. Adams, E. Chaubey and S. Weinzierl, Planar Double Box Integral for Top Pair Production with a Closed Top Loop to all orders in the Dimensional Regularization Parameter, Phys. Rev. Lett. 121 (2018) 142001 [arXiv:1804.11144] [INSPIRE].ADSCrossRefGoogle Scholar
  128. [128]
    L. Adams, E. Chaubey and S. Weinzierl, Analytic results for the planar double box integral relevant to top-pair production with a closed top loop, JHEP 10 (2018) 206 [arXiv:1806.04981] [INSPIRE].ADSCrossRefGoogle Scholar
  129. [129]
    J. Broedel, C. Duhr, F. Dulat, B. Penante and L. Tancredi, From modular forms to differential equations for Feynman integrals, in proceedings of the KMPB Conference: Elliptic Integrals, Elliptic Functions and Modular Forms in Quantum Field Theory, Zeuthen, Germany, 23-26 October 2017, arXiv:1807.00842 [INSPIRE].
  130. [130]
    L. Adams and S. Weinzierl, On a class of Feynman integrals evaluating to iterated integrals of modular forms, in proceedings of the KMPB Conference: Elliptic Integrals, Elliptic Functions and Modular Forms in Quantum Field Theory, Zeuthen, Germany, 23-26 October 2017, arXiv:1807.01007 [INSPIRE].
  131. [131]
    J. Blümlein, A. De Freitas, M. Van Hoeij, E. Imamoglu, P. Marquard and C. Schneider, The ρ parameter at three loops and elliptic integrals, in proceedings of the 14th DESY Workshop on Elementary Particle Physics: Loops and Legs in Quantum Field Theory 2018 (LL2018), St Goar, Germany, 29 April-4 May 2018 [PoS(LL2018)017 (2018)] [arXiv:1807.05287] [INSPIRE].
  132. [132]
    J. Blümlein, Iterative Non-iterative Integrals in Quantum Field Theory, in proceedings of the KMPB Conference: Elliptic Integrals, Elliptic Functions and Modular Forms in Quantum Field Theory, Zeuthen, Germany, 23-26 October 2017, arXiv:1808.08128 [INSPIRE].
  133. [133]
    P. Vanhove, Feynman integrals, toric geometry and mirror symmetry, in proceedings of the KMPB Conference: Elliptic Integrals, Elliptic Functions and Modular Forms in Quantum Field Theory, Zeuthen, Germany, 23-26 October 2017, arXiv:1807.11466 [INSPIRE].
  134. [134]
    J.L. Bourjaily, Y.-H. He, A.J. Mcleod, M. Von Hippel and M. Wilhelm, Traintracks through Calabi-Yau Manifolds: Scattering Amplitudes beyond Elliptic Polylogarithms, Phys. Rev. Lett. 121 (2018) 071603 [arXiv:1805.09326] [INSPIRE].
  135. [135]
    F.C.S. Brown and O. Schnetz, A K3 in \( \phi \) 4, Duke Math. J. 161 (2012) 1817 [arXiv:1006.4064] [INSPIRE].
  136. [136]
    S. Bloch, M. Kerr and P. Vanhove, A Feynman integral via higher normal functions, Compos. Math. 151 (2015) 2329 [arXiv:1406.2664] [INSPIRE].MathSciNetCrossRefzbMATHGoogle Scholar
  137. [137]
    F. Brown and A. Levin, Multiple Elliptic Polylogarithms, arXiv:1110.6917.
  138. [138]
    J. Broedel, C.R. Mafra, N. Matthes and O. Schlotterer, Elliptic multiple zeta values and one-loop superstring amplitudes, JHEP 07 (2015) 112 [arXiv:1412.5535] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  139. [139]
    J. Broedel, N. Matthes and O. Schlotterer, Relations between elliptic multiple zeta values and a special derivation algebra, J. Phys. A 49 (2016) 155203 [arXiv:1507.02254] [INSPIRE].
  140. [140]
    J. Broedel, N. Matthes, G. Richter and O. Schlotterer, Twisted elliptic multiple zeta values and non-planar one-loop open-string amplitudes, J. Phys. A 51 (2018) 285401 [arXiv:1704.03449] [INSPIRE].
  141. [141]
    J. Broedel, O. Schlotterer and F. Zerbini, From elliptic multiple zeta values to modular graph functions: open and closed strings at one loop, arXiv:1803.00527 [INSPIRE].
  142. [142]
    O. Schlotterer and O. Schnetz, Closed strings as single-valued open strings: A genus-zero derivation, arXiv:1808.00713 [INSPIRE].
  143. [143]
    Y.I. Manin, Iterated integrals of modular forms and noncommutative modular symbols, in Algebraic geometry and number theory, Progress in Mathematics, volume 253, Birkhäuser Boston U.S.A. (2006), pp. 565-597 [math.NT/0502576].
  144. [144]
    F.C.S. Brown, Multiple modular values and the relative completion of the fundamental group of1,1, arXiv:1407.5167v4.
  145. [145]
    L. Adams and S. Weinzierl, Feynman integrals and iterated integrals of modular forms, Commun. Num. Theor. Phys. 12 (2018) 193 [arXiv:1704.08895] [INSPIRE].MathSciNetCrossRefzbMATHGoogle Scholar
  146. [146]
    L. Adams and S. Weinzierl, The ε-form of the differential equations for Feynman integrals in the elliptic case, Phys. Lett. B 781 (2018) 270 [arXiv:1802.05020] [INSPIRE].
  147. [147]
    J. Broedel, C. Duhr, F. Dulat, B. Penante and L. Tancredi, Feynman parametric integrals and elliptic polylogarithms, to appear.Google Scholar
  148. [148]
    J.A. Lappo-Danilevsky, Théorie algorithmique des corps de Riemann, Rec. Math. Moscou 34 (1927) 113.Google Scholar
  149. [149]
    A.B. Goncharov, Multiple polylogarithms and mixed Tate motives, math.AG/0103059 [INSPIRE].
  150. [150]
    F. Cachazo, Sharpening The Leading Singularity, arXiv:0803.1988 [INSPIRE].
  151. [151]
    R.N. Lee, Reducing differential equations for multiloop master integrals, JHEP 04 (2015) 108 [arXiv:1411.0911] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  152. [152]
    O. Gituliar and V. Magerya, Fuchsia: a tool for reducing differential equations for Feynman master integrals to epsilon form, Comput. Phys. Commun. 219 (2017) 329 [arXiv:1701.04269] [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  153. [153]
    C. Meyer, Transforming differential equations of multi-loop Feynman integrals into canonical form, JHEP 04 (2017) 006 [arXiv:1611.01087] [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  154. [154]
    K.G. Chetyrkin and F.V. Tkachov, Integration by Parts: The Algorithm to Calculate β-functions in 4 Loops, Nucl. Phys. B 192 (1981) 159 [INSPIRE].
  155. [155]
    F.V. Tkachov, A Theorem on Analytical Calculability of Four Loop Renormalization Group Functions, Phys. Lett. B 100 (1981) 65 [INSPIRE].
  156. [156]
    N. Matthes, Elliptic Multiple Zeta Values, Ph.D. Thesis, Universität Hamburg (2016).Google Scholar
  157. [157]
    F.C.S. Brown, Notes on Motivic Periods, Commun. Num. Theor. Phys. 11 (2017) 557 [arXiv:1512.06410].MathSciNetCrossRefzbMATHGoogle Scholar
  158. [158]
    A.B. Goncharov, Galois symmetries of fundamental groupoids and noncommutative geometry, Duke Math. J. 128 (2005) 209 [math.AG/0208144] [INSPIRE].
  159. [159]
    F.C.S. Brown, On the decomposition of motivic multiple zeta values, in Galois-Teichmüller theory and arithmetic geometry, Advanced Studies in Pure Mathematics, volume 68, Mathematical Society of Japan, Tokyo Japan (2012), pp. 31-58 [arXiv:1102.1310] [INSPIRE].
  160. [160]
    D. Zagier, private communication (2018).Google Scholar
  161. [161]
    L. Adams, C. Bogner and S. Weinzierl, The two-loop sunrise graph with arbitrary masses, J. Math. Phys. 54 (2013) 052303 [arXiv:1302.7004] [INSPIRE].
  162. [162]
    E. Remiddi and L. Tancredi, An Elliptic Generalization of Multiple Polylogarithms, Nucl. Phys. B 925 (2017) 212 [arXiv:1709.03622] [INSPIRE].
  163. [163]
    R.J. Gonsalves, Dimensionally Regularized Two Loop On-Shell Quark Form-Factor, Phys. Rev. D 28 (1983) 1542 [INSPIRE].
  164. [164]
    W.L. van Neerven, Dimensional Regularization of Mass and Infrared Singularities in Two Loop On-shell Vertex Functions, Nucl. Phys. B 268 (1986) 453 [INSPIRE].
  165. [165]
    G. Kramer and B. Lampe, Integrals for Two Loop Calculations in Massless QCD, J. Math. Phys. 28 (1987) 945 [INSPIRE].
  166. [166]
    T. Gehrmann, T. Huber and D. Maître, Two-loop quark and gluon form-factors in dimensional regularisation, Phys. Lett. B 622 (2005) 295 [hep-ph/0507061] [INSPIRE].
  167. [167]
    C. Bogner and S. Weinzierl, Periods and Feynman integrals, J. Math. Phys. 50 (2009) 042302 [arXiv:0711.4863] [INSPIRE].
  168. [168]
    S. Zemel, A Direct Evaluation of the Periods of the Weierstrass Zeta Function, Ann. Univ. Ferrara 60 (2014) 495 [arXiv:1304.7194].MathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© The Author(s) 2019

Authors and Affiliations

  • Johannes Broedel
    • 1
  • Claude Duhr
    • 2
    • 3
  • Falko Dulat
    • 4
  • Brenda Penante
    • 2
  • Lorenzo Tancredi
    • 2
    Email author
  1. 1.Institut für Mathematik und Institut für PhysikHumboldt-Universität zu BerlinBerlinGermany
  2. 2.Theoretical Physics DepartmentCERNGenevaSwitzerland
  3. 3.Center for Cosmology, Particle Physics and Phenomenology (CP3)Universiteé Catholique de LouvainLouvain-La-NeuveBelgium
  4. 4.SLAC National Accelerator LaboratoryStanford UniversityMenlo ParkU.S.A.

Personalised recommendations